Development of Equations of Motion for a Stewart Platform Under Prescribed Mount Motion

2018 ◽  
Author(s):  
Takeyuki Ono ◽  
Ryosuke Eto ◽  
Junya Yamakawa ◽  
Hidenori Murakami
Keyword(s):  
Rigid Body ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Base Plate ◽  
Stewart Platform ◽  
Euler Method ◽  
Moving Frame ◽  
Real Time Control ◽  
Precise Positioning ◽  
Time Control

Analytical equations of motion are critical for real-time control of translating manipulators, which require precise positioning of various tools for their mission. Specifically, when manipulators mounted on moving robots or vehicles perform precise positioning of their tools, it becomes economical to develop a Stewart platform, whose sole task is stabilizing the orientation and crude position of its top table, onto which various precision tools are attached. In this paper, analytical equations of motion are developed for a Stewart platform whose motion of the base plate is prescribed. To describe the kinematics of the platform, the moving frame method, presented by one of authors [1,2], is employed. In the method the coordinates of the origin of a body attached coordinate system and vector basis are expressed by using 4 × 4 frame connection matrices, which form the special Euclidean group, SE(3). The use of SE(3) allows accurate description of kinematics of each rigid body using (relative) joint coordinates. In kinetics, the principle of virtual work is employed, in which system virtual displacements are expressed through B-matrix by essential virtual displacements, reflecting the connection of the rigid body system [2]. The resulting equations for fixed base plate reduce to those for the top plate, obtained by the Newton-Euler method. A main result of the paper is the analytical equations of motion in matrix form for dynamics analyses of a Stewart platform whose base plate moves. The control applications of those equations will be deferred to subsequent publications.

scholarly journals Kinematic support modeling in Sage

2020 ◽  
Vol 28 (2) ◽  
pp. 141-153
Author(s):  
Oleg K. Kroytor ◽  
Mikhail D. Malykh ◽  
Sergei P. Karnilovich
Keyword(s):  
Rigid Body ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Seismic Isolation ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Vertical Plane ◽  
Base Plate ◽  
Euler Method ◽  
Point Of View

The article discusses the kinematic support, which allows reducing the horizontal dynamic effects on the building during earthquakes. The model of a seismic isolation support is considered from the point of view of classical mechanics, that is, we assume that the support is absolutely solid, oscillating in a vertical plane above a fixed horizontal solid plate. This approach allows a more adequate description of the interaction of the support with the soil and the base plate of the building. The paper describes the procedure for reducing the complete system of equations of motion of a massive rigid body on a fixed horizontal perfectly smooth plane to a form suitable for applying the finite difference method and its implementation in the Sage computer algebra system. The numerical calculations by the Euler method for grids with different number of elements are carried out and a mathematical model of the support as a perfectly rigid body in the Sage computer algebra system is implemented. The article presents the intermediate results of numerical experiments performed in Sage and gives a brief analysis (description) of the results.

RESEARCH ON RIGID BODY INVERSE DYNAMICS OF A NOVEL 6-PRRS PARALLEL ROBOT

2018 ◽  
Vol 18 (08) ◽  
pp. 1840037
Author(s):  
YUBIN LIU ◽  
GANGFENG LIU
Keyword(s):  
Dynamic Model ◽  
Inverse Dynamics ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Parallel Robot ◽  
Simulation Software ◽  
Real Time Control ◽  
Dynamical Equations ◽  
Time Control ◽  
Systematic Methodology

A systematic methodology for solving the inverse dynamics of a 6-PRRS parallel robot is presented. Based on the principle of virtual work and the Lagrange approach, a methodology for deriving the dynamical equations of motion is developed. To resolve the inconsistency between complications of established dynamic model and real-time control, a simplifying strategy of the dynamic model is presented. The dynamic character of the 6-PRRS parallel robot is analyzed by example calculation, and a full and precise dynamic model using simulation software is established. Verification results show the validity of the presented algorithm, and the simplifying strategies are practical and efficient.

Dynamic Modeling and Analysis for a Planar Three Degrees-of-Freedom Manipulator Under Prescribed Mount Motion

2019 ◽  
Author(s):  
Takeyuki Ono ◽  
Ryosuke Eto ◽  
Junya Yamakawa ◽  
Hidenori Murakami
Keyword(s):  
Dynamic Modeling ◽  
Degrees Of Freedom ◽  
Sliding Mode ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Base Plate ◽  
Moving Frame ◽  
Parallel Plates ◽  
Modeling And Analysis ◽  
Three Degrees Of Freedom

Abstract This paper presents dynamic modeling of a planar, three degrees-of-freedom manipulator consisting of two parallel plates, referred to as top and base plates, which are connected by three actuated legs. When a sensitive equipment is carried by a moving robot or vehicle, it becomes necessary to mount the equipment on a platform which achieves precise positioning for stabilization. The objectives of this paper are to derive analytical equations of motion and apply them to control simulations on the stabilizing planar manipulator. In the derivation of analytical equations of motion, the moving frame method is utilized to describe the kinematics of the two-dimensional multibody system. For the manipulator system comprised of jointed bodies, a graph tree is utilized, which visually illustrates how the constituent bodies are connected to each other. For kinetics, the principle of virtual work is employed to derive the analytical equations of motion for the manipulator system. The resulting equations of motion are used to numerically assess the performance of a sliding mode controller (SMC) to stabilize the top plate from the motion of the translating and rotating base plate. In the numerical simulation, the SMC is compared with a simple PID controller to evaluate both the tracking performance and robustness.

Dynamic Model of a Spur Gear System With Backlash and Friction Consideration

1994 ◽  
Author(s):  
T. K. Shing ◽  
Lung-Wen Tsai ◽  
P. S. Krishnaprasad
Keyword(s):  
Free Oscillation ◽  
Equations Of Motion ◽  
Spur Gear ◽  
Constant Load ◽  
Friction Torque ◽  
Gear System ◽  
Real Time Control ◽  
Dynamical Equations ◽  
Gear Noise ◽  
Time Control

Abstract A new model which accounts for both backlash and friction effects is proposed for the dynamics of a spur gear system. The model estimates average friction torque and uses it to replace the instantaneous friction torque to simplify the dynamical equations of motion. Two simulations, free oscillation and constant load operation, are performed to illustrate the effects of backlash and friction on gear dynamics. The results are compared with that of a previously established model which does not account for the friction. Finally, the effect of adding a damper on the driving shaft is also studied. This model is judged to be more realistic for real time control of electronmechanical systems to reduce gear noise and to achieve high precision.

Control Simulations for a Stewart Platform Compensator Mounted on Moving Base

2019 ◽  
Author(s):  
Takeyuki Ono ◽  
Ryosuke Eto ◽  
Junya Yamakawa ◽  
Hidenori Murakami
Keyword(s):  
Time Delay ◽  
Sliding Mode ◽  
Equations Of Motion ◽  
Base Plate ◽  
Medical Application ◽  
Stewart Platform ◽  
Hybrid Controller ◽  
Resolved Acceleration Control ◽  
Moving Base ◽  
Input Time

Abstract In this paper, utilizing the analytical equations of motion for a base-moving Stewart platform, we design an active wave compensation system for a surgery table installed on the top plate of a Stewart platform in a ship. In our medical application, the base plate of a Stewart platform moves with the motion of the ship. For a base-moving Stewart platform, we presented analytical equations of motion in matrix form in the paper: IMECE2018-87253. The objective of the platform is to compensate the pitching, rolling, and heaving motions of the ship (with respect to an inertial coordinate system). As control methods for the nonlinear system, we employ a hybrid controller combining resolved acceleration control with H∞ control, and integral sliding mode control (ISMC). The ISMC with input time delay is also designed with a state predictor, which includes a ship motion predictor utilizing an autoregressive model. Finally, to assess the control performance and robustness for the system with uncertainties, numerical simulations are presented. In addition, the simulation results of the predictor based ISMC for the system with input time delay are illustrated showing the effectiveness of the controller.

Kinematics optimization and static analysis of a modular continuum robot used for minimally invasive surgery

2017 ◽  
Vol 232 (2) ◽  
pp. 135-148 ◽  
Author(s):  
Fei Qi ◽  
Feng Ju ◽  
Dong Ming Bai ◽  
Bai Chen
Keyword(s):  
Minimally Invasive Surgery ◽  
Minimally Invasive ◽  
Invasive Surgery ◽  
Virtual Work ◽  
Structural Parameters ◽  
Real Time Control ◽  
Time Control ◽  
Payload Capacity ◽  
Continuum Robot ◽  
The Relationship

For the outstanding compliance and dexterity of continuum robot, it is increasingly used in minimally invasive surgery. The wide workspace, high dexterity and strong payload capacity are essential to the continuum robot. In this article, we investigate the workspace of a cable-driven continuum robot that we proposed. The influence of section number on the workspace is discussed when robot is operated in narrow environment. Meanwhile, the structural parameters of this continuum robot are optimized to achieve better kinematic performance. Moreover, an indicator based on the dexterous solid angle for evaluating the dexterity of robot is introduced and the distal end dexterity is compared for the three-section continuum robot with different range of variables. Results imply that the wider range of variables achieve the better dexterity. Finally, the static model of robot based on the principle of virtual work is derived to analyze the relationship between the bending shape deformation and the driven force. The simulations and experiments for plane and spatial motions are conducted to validate the feasibility of model, respectively. Results of this article can contribute to the real-time control and movement and can be a design reference for cable-driven continuum robot.

A New Approach to Model Constant Curvature Continuum Robot Dynamics

2019 ◽  
Author(s):  
Yujiong Liu ◽  
Pinhas Ben-Tzvi
Keyword(s):  
Constant Curvature ◽  
Inverse Dynamics ◽  
Virtual Work ◽  
Robot Dynamics ◽  
Real Time Control ◽  
Continuum Robots ◽  
New Approach ◽  
Time Control ◽  
Proposed Model ◽  
Vector Representations

Abstract Inspired by nature, continuum robots show their potential in human-centered environments due to the compliant-to-obstacle features and dexterous mobility. However, there are few such robots successfully implemented outside the laboratory so far. One reason is believed to be due to the real time control challenge for soft robots, which require a highly efficient, highly accurate dynamic model. This paper presents a new systematic methodology to formulate the dynamics of constant curvature continuum robots. The new approach builds on several new techniques: 1) using the virtual work principle to formulate the equation of motion, 2) using specifically selected kinematic representations to separate integral variables from the non-integral variables, and 3) using vector representations to put the integral in a compact form. By doing so, the hard-to-solve integrals are evaluated analytically in advance and the accurate inverse dynamics are established accordingly. Numerical simulations are conducted to evaluate the performances of the newly proposed model.

Development of an Active Curved Beam Model—Part I: Kinematics and Integrability Conditions

2017 ◽  
Vol 84 (6) ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Critical Role ◽  
Three Dimensional ◽  
Moving Frame ◽  
Beam Model ◽  
Mathematical Tool ◽  
Differentiable Manifolds ◽  
Beam Equations ◽  
Integrability Conditions

In order to develop an active nonlinear beam model, the beam's kinematics is examined in this paper, by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. These integrability conditions enable the derivation of beam models in Part II, starting from the three-dimensional Hamilton's principle and the d'Alembert's principle of virtual work. To illustrate the critical role played by the integrability conditions, the variation of kinetic energy is computed. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.

Integrability Conditions in Nonlinear Beam Kinematics

2016 ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Moving Frame ◽  
Beam Model ◽  
Mathematical Tool ◽  
Nonlinear Beam ◽  
Differentiable Manifolds ◽  
Beam Equations ◽  
Integrability Conditions

In order to develop an active nonlinear beam model, the beam’s kinematics is examined by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Élie Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. They also serve a role in a geometrically-exact finite-element implementation of beam models. These integrability conditions enable the derivation of beam models starting from the three-dimensional Hamilton’s principle and the d’Alembert principle of virtual work. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.

Gyroscopic Wave Energy Generator for Fish Farms and Rigs

2018 ◽  
Author(s):  
Alexandra Norbach ◽  
Kotryna Bedrovaite Fjetland ◽  
Gina Vikum Hestetun ◽  
Thomas J. Impelluso
Keyword(s):  
Wave Energy ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Integration Scheme ◽  
Moving Frames ◽  
Fish Farms ◽  
Alternative Source ◽  
Ocean Wave Energy ◽  
Angular Velocities

Norway has an opportunity to harvest ocean wave energy through gyroscopic precession as an alternative source of renewable energy, within practical limitations. This research assesses the energy extracted by gyroscopic wave energy generators and their use to provide supplementary power to fish farms and lighting on oilrigs. This project implements the Moving Frame Method (MFM) in dynamics to model the extracted power from a gyroscopic wave energy generator. The MFM leverages Lie Group Theory, Cartan’s moving frames and a new notation from the discipline of geometrical physics. Continuing, the Principle of Virtual Work extracts the equations of motion from the structure of the Special Orthogonal Group. However, the MFM supplements its analysis with a novel application of the restriction on the variations of the angular velocities. This research extends previous work as follows: it accounts for motor torques, it opens a placeholder for buoyancy, and it solves the full 3D set of equations (without assuming negligible yaw). After showing how to obtain the suite of descriptive equations of motion, this project integrates them, however with a relatively simple integration scheme. To complete each step in the analysis, the rotation matrix is updated using the Cayley Hamilton Theorem and the Rodriguez formula. Finally, the results are displayed using the Web Graphics Library such that the actual numerical analysis and display happens on cell phones.

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